5 EXAMPLES OF ANALYTIC FUNCTIONS 213

mapping of the restriction of x -* eix to ]a, a + 2n[ was not continuous
at Co» there would be in ]a, a + 2n[ a sequence (xn) whose elements would
belong to the complement of a neighborhood of 6, and such that
lim eiXn = Co; but then a subsequence (xnj) would tend to a limit c ^ b

n-> oo

in the compact set [a, a + 2n] by (3.16.1), and as eic *£ eib we arrive at a
contradiction. (For another proof, see (10.3.1).)

(9.5.8) The unit circle U is connected.

This follows from (9.5.7), (3.19.1) and (3.19.7).

(9.5.9) ("Principle of maximum") Let (cvzv) be a power series with
complex coefficients, absolutely summable in an open poly disk P a Cp of center 0
and let /(z) be its sum. Suppose that there is an open ball B c P of center 0
such that |/(z)| < |/(0)| for every ZGB. Then cv = 0 for every index
v T£ (0, . . . , 0), in other words, f is a constant.

We first prove that the theorem is true for any p if it is true for p = 1 .
Indeed, for any z = (z1? . . . , zp) e P, consider the function of one complex
variable g(i) =f(tzl9 . . . , tzp) which is analytic for \t\ < 1 + & with e small
enough. As \g(t)\ < \g(G)\ for small values of t, we have g(t)=g(Q) by
assumption, and in particular /(zt..., zp) =#(1) =/(0). For p=l, we
can suppose c0 ^ 0, otherwise the result is obvious by (9.1.6). Suppose
there are indices n > 0 such that cn ^ 0, and let m be the smallest of them.
We can write

where bm^0,h is analytic in P and /z(0) = 0. Let r > 0 be such that \z\ ^ r
is contained in B and \h(z)\ <%\bm\ for \z\ <r (9.1.3). Write bm = \bm\£
with Id = 1 ; by (9.5.7) there is a real t such that emit = C"1; for z = relt,
we therefore have

[I + bmzm + zw/z(z)| = |1 + \bm\rm

which contradicts the assumption |/(z)| ^ |c0| in B.

The result (9,5.9) does not hold if Cp is replaced by Rp, as the example

of the power series 1/(1 + z2) = £ (-I)nz2n (for |z| < 1) shows.ion e* of the real variable x; we prove it is an